Abstract

Pulse propagation in a random medium is determined by the two-frequency mutual coherence function which satisfies a parabolic equation. In the past, numerical solutions of this equation have been reported for the plane wave case. An exact analytical solution for the plane wave case has also been reported for a Gaussian spectrum of refractive-index fluctuations. Using the same approximation, an exact analytic solution for the more general case of an incident beam wave is presented. The solution so obtained is used to study the propagation characteristics of the beam wave mutual coherence function at a single frequency as well as at two frequencies. Simple expressions are obtained which qualitatively describe the decollimating and defocusing effects of turbulence on a propagating beam wave. The time variation of the received pulse shape, on and away from the beam axis, is studied when the medium is excited with a delta function input. The results are presented for both collimated and focused beams.

© 1979 Optical Society of America

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References

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  1. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Springfield, Va., 1971).
  2. Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tatarski, Sov. Phys. Usp. 13, 551 (1971).
    [CrossRef]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vols. 1 and 2 (Academic, New York, 1978).
  4. C. H. Liu, A. W. Wernik, K. C. Yeh, M. Y. Youakim, Radio Sci., 9, 599 (1974).
    [CrossRef]
  5. L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).
    [CrossRef]
  6. L. C. Lee, J. Math. Phys. 15, 1431 (1974).
    [CrossRef]
  7. S. T. Hong, I. Sreenivasiah, A. Ishimaru, IEEE Trans. Antennas Propag. AP-25, 822 (1976).
  8. I. Sreenivasiah, A. Ishimaru, “Plane wavepulse propagation through atmospheric turbulence at millimeter and optical wavelengths,” Department of Electrical Engineering, University of Washington, Research Report AFCRL-TR-74-0205 (1974).
  9. I. Sreenivasiah, “Two-frequency mutual coherence function and pulse propagation in continuous random media: Forward and backscattering solutions,” Ph.D. Dissertation, University of Washington, Seattle (1976).
  10. I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1976).
    [CrossRef]
  11. L. Erukhimov, I. Zarnitsyna, P. Kirsch, Izv. VUZ. Radiofizika 16, 573 (1973).

1976

S. T. Hong, I. Sreenivasiah, A. Ishimaru, IEEE Trans. Antennas Propag. AP-25, 822 (1976).

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1976).
[CrossRef]

1975

L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).
[CrossRef]

1974

L. C. Lee, J. Math. Phys. 15, 1431 (1974).
[CrossRef]

C. H. Liu, A. W. Wernik, K. C. Yeh, M. Y. Youakim, Radio Sci., 9, 599 (1974).
[CrossRef]

1973

L. Erukhimov, I. Zarnitsyna, P. Kirsch, Izv. VUZ. Radiofizika 16, 573 (1973).

1971

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tatarski, Sov. Phys. Usp. 13, 551 (1971).
[CrossRef]

Barabanenkov, Y. N.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tatarski, Sov. Phys. Usp. 13, 551 (1971).
[CrossRef]

Erukhimov, L.

L. Erukhimov, I. Zarnitsyna, P. Kirsch, Izv. VUZ. Radiofizika 16, 573 (1973).

Hong, S. T.

S. T. Hong, I. Sreenivasiah, A. Ishimaru, IEEE Trans. Antennas Propag. AP-25, 822 (1976).

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1976).
[CrossRef]

Ishimaru, A.

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1976).
[CrossRef]

S. T. Hong, I. Sreenivasiah, A. Ishimaru, IEEE Trans. Antennas Propag. AP-25, 822 (1976).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vols. 1 and 2 (Academic, New York, 1978).

I. Sreenivasiah, A. Ishimaru, “Plane wavepulse propagation through atmospheric turbulence at millimeter and optical wavelengths,” Department of Electrical Engineering, University of Washington, Research Report AFCRL-TR-74-0205 (1974).

Jokipii, J. R.

L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).
[CrossRef]

Kirsch, P.

L. Erukhimov, I. Zarnitsyna, P. Kirsch, Izv. VUZ. Radiofizika 16, 573 (1973).

Kravtsov, Y. A.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tatarski, Sov. Phys. Usp. 13, 551 (1971).
[CrossRef]

Lee, L. C.

L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).
[CrossRef]

L. C. Lee, J. Math. Phys. 15, 1431 (1974).
[CrossRef]

Liu, C. H.

C. H. Liu, A. W. Wernik, K. C. Yeh, M. Y. Youakim, Radio Sci., 9, 599 (1974).
[CrossRef]

Rytov, S. M.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tatarski, Sov. Phys. Usp. 13, 551 (1971).
[CrossRef]

Sreenivasiah, I.

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1976).
[CrossRef]

S. T. Hong, I. Sreenivasiah, A. Ishimaru, IEEE Trans. Antennas Propag. AP-25, 822 (1976).

I. Sreenivasiah, “Two-frequency mutual coherence function and pulse propagation in continuous random media: Forward and backscattering solutions,” Ph.D. Dissertation, University of Washington, Seattle (1976).

I. Sreenivasiah, A. Ishimaru, “Plane wavepulse propagation through atmospheric turbulence at millimeter and optical wavelengths,” Department of Electrical Engineering, University of Washington, Research Report AFCRL-TR-74-0205 (1974).

Tatarski, V. I.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tatarski, Sov. Phys. Usp. 13, 551 (1971).
[CrossRef]

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Springfield, Va., 1971).

Wernik, A. W.

C. H. Liu, A. W. Wernik, K. C. Yeh, M. Y. Youakim, Radio Sci., 9, 599 (1974).
[CrossRef]

Yeh, K. C.

C. H. Liu, A. W. Wernik, K. C. Yeh, M. Y. Youakim, Radio Sci., 9, 599 (1974).
[CrossRef]

Youakim, M. Y.

C. H. Liu, A. W. Wernik, K. C. Yeh, M. Y. Youakim, Radio Sci., 9, 599 (1974).
[CrossRef]

Zarnitsyna, I.

L. Erukhimov, I. Zarnitsyna, P. Kirsch, Izv. VUZ. Radiofizika 16, 573 (1973).

Astrophys. J.

L. C. Lee, J. R. Jokipii, Astrophys. J. 201, 532 (1975).
[CrossRef]

IEEE Trans. Antennas Propag.

S. T. Hong, I. Sreenivasiah, A. Ishimaru, IEEE Trans. Antennas Propag. AP-25, 822 (1976).

Izv. VUZ. Radiofizika

L. Erukhimov, I. Zarnitsyna, P. Kirsch, Izv. VUZ. Radiofizika 16, 573 (1973).

J. Math. Phys.

L. C. Lee, J. Math. Phys. 15, 1431 (1974).
[CrossRef]

Radio Sci.

C. H. Liu, A. W. Wernik, K. C. Yeh, M. Y. Youakim, Radio Sci., 9, 599 (1974).
[CrossRef]

I. Sreenivasiah, A. Ishimaru, S. T. Hong, Radio Sci. 11, 775 (1976).
[CrossRef]

Sov. Phys. Usp.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tatarski, Sov. Phys. Usp. 13, 551 (1971).
[CrossRef]

Other

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vols. 1 and 2 (Academic, New York, 1978).

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Springfield, Va., 1971).

I. Sreenivasiah, A. Ishimaru, “Plane wavepulse propagation through atmospheric turbulence at millimeter and optical wavelengths,” Department of Electrical Engineering, University of Washington, Research Report AFCRL-TR-74-0205 (1974).

I. Sreenivasiah, “Two-frequency mutual coherence function and pulse propagation in continuous random media: Forward and backscattering solutions,” Ph.D. Dissertation, University of Washington, Seattle (1976).

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Figures (4)

Fig. 1
Fig. 1

Maximum distance of collimation in the presence of turbulence. zmax vs W0.

Fig. 2
Fig. 2

Γ1(ρc = ρd = 0,η) vs η for a collimated beam. λ = 0.6943 μm; z = 10 km and n 1 2 / l = 10 - 14 m - 1 . W0 1 = mm, 1 cm, 3.5 cm, 10 cm, 25 cm, and ≥1 m.

Fig. 3
Fig. 3

Γ1(ρc,ρd = 0,η)/Γ1(ρc,ρd = 0,η = 0) vs η. λ = 0.6943 μm; z = 10 km and n 1 2 / l = 10 - 14 m - 1 . (a) Collimated beam: W0 = 10 cm; ρc = 0, 5 cm, 10 cm, and 15 cm. W0 =25 cm; ρc = 0 and 15 cm. (b) Focused beam: W0 = 10 cm and 25 cm; ρc = 0, 5 cm, 10 cm, and 15 cm.

Fig. 4
Fig. 4

In(ρc,tn) = I1(ρc,tnT1 vs tn = t - [z/(v0]/T1. λ = 0.6943 μm; z = 10 km and n 1 2 / l = 10 - 14 m - 1 . (a) Collimated beam: W0 = 10 cm and 25 cm; ρc = 0. 5 cm, 10 cm, and 15 cm. (b) Focused beam: W0 = 10 cm and 25 cm; ρc = 0, 5 cm, and 10 cm.

Equations (50)

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r ( r ¯ , ω , t ) = r ( r ¯ , ω , t ) [ 1 + 1 ( r ¯ , ω , t ) ] .
E ( r ¯ , ω , t ) = u ( z , ρ ¯ , ω , t ) exp [ i ( k z - ω t ) ] ,
( 2 i z + 1 k 1 1 2 - 1 k 2 2 2 + i 4 { k 1 2 A 1 ( 0 ) + k 2 2 A 2 ( 0 ) - k 1 k 2 [ A 1 ( ρ ¯ d - V ¯ t d ) + A 2 ( ρ ¯ d - V ¯ t d ) ] } ) Γ = 0 ,
A 1 ( ρ ¯ ) = 16 π 2 0 J 0 ( κ ρ ) Φ n 1 ( κ ) κ d κ .
D ( ρ ) = A ( 0 ) - A ( ρ ) ρ 2 .
Φ n ( κ ) = n 1 2 l 3 8 π π exp ( - 1 4 κ 2 l 2 ) ,
D ( ρ ) = 4 π n 1 2 l [ 1 - exp ( - ρ 2 l 2 ) ] C ρ 2 ,
C = 4 π n 1 2 l - 1 = [ A ( 0 ) ] / l 2 .
Γ 2 = exp [ - 1 8 k d 2 A ( 0 ) z ] ,
( z + β d 2 + β 4 d 2 + i γ ¯ d · ¯ c + δ ρ d 2 ) Γ 1 ( z , ρ ¯ c , ρ ¯ d , k c , k d ) = 0 ,
β = ( i k d ) / ( 2 k 1 k 2 ) , γ = [ k c / ( k 1 k 2 ) ] , δ = [ ( k 1 k 2 ) / 4 ] { [ A ( 0 ) ] / l 2 } , ρ ¯ d = ρ ¯ 1 - ρ ¯ 2 , ρ ¯ c = ½ ( ρ ¯ 1 + ρ ¯ 2 ) , k d = k 1 - k 2 , k c = ½ ( k 1 + k 2 )
E ( z , ρ ¯ , k , t ) = u ( z , ρ ¯ , k ) exp { i k [ z - ( t / v 0 ) ] } ,
u ( z , ρ ¯ , k ) = 1 1 + i α z exp ( - k α 2 ρ 2 1 + i α z ) , α = λ π W 0 2 + i 1 R 0 = α r + i α i .
Γ ( z = 0 ) = Γ ( 0 , ρ ¯ c , ρ ¯ d , k c , k d ) = G 0 exp ( - a 0 ρ d 2 - b 0 ρ c 2 - i c 0 ρ ¯ c · ρ ¯ d ) ,
a 0 = k c α r 4 + i k d 8 α i , b 0 = 4 a 0 , c 0 = k c α i - i k d 2 α r ,
G 0 = 1 π 2 W 0 4 ( 1 + k c 2 R 0 2 W 0 4 4 ) .
I 1 ( z , κ ¯ d , ρ ¯ d , k c , k d ) = 1 ( 2 π ) 2 Γ 1 exp ( - i κ ¯ d · ρ ¯ c ) d ρ ¯ c , Γ 1 ( z , ρ ¯ c , ρ ¯ d , k c , k d ) = I 1 exp ( i κ ¯ d · ρ ¯ c ) d κ ¯ d .
( z + β d 2 - β 4 κ d 2 - γ κ ¯ d · ¯ d + δ ρ d 2 ) I 1 = 0 ,
I 1 ( z = 0 ) = I 0 = G 0 exp ( - a 0 ρ d 2 - b 0 κ d 2 - c 0 κ ¯ d · ρ ¯ d ) ,
G 0 = G 0 / ( 4 π b 0 ) a 0 = [ ( c 0 2 ) / ( 4 b 0 ) ] + a 0 b 0 = 1 / ( 4 b 0 ) c 0 = c 0 / ( 2 b 0 ) } .
I 1 = G f exp ( - g 1 ρ d 2 - g 2 κ d 2 - g 3 κ ¯ d · ρ ¯ d ) ,
[ ( f ) / ( z ) ] - 4 β g 1 f = 0 ,
[ ( g 1 ) / ( z ) ] - 4 β g 1 2 - δ = 0 ,
[ ( g 2 ) / ( z ) ] - β g 3 2 - γ g 3 + ( β / 4 ) = 0 ,
[ ( g 3 ) / ( z ) ] - 4 β g 1 g 3 - 2 γ g 1 = 0 ,
G = [ ( G 0 b 0 ) / ( π f 0 ) ] f 0 = f ( z = 0 ) g 10 = g 1 ( z = 0 ) = a 0 g 20 = g 2 ( z = 0 ) = b 0 g 30 = g 3 ( z = 0 ) = c 0 } .
Γ 1 = G 0 ( g 20 g 2 ) ( f f 0 ) exp ( - a ρ d 2 - b ρ c 2 - i c ρ ¯ c · ρ ¯ d ) ,
a = g 1 - [ ( g 3 2 ) / ( 4 g 2 ) ] , b = 1 / ( 4 g 2 ) , c = g 3 / ( 2 g 2 ) , f = sec [ ( 4 β δ ) 1 / 2 z + θ 0 ] , g 1 = ( δ 4 β ) 1 / 2 tan [ ( 4 β δ ) 1 / 2 z + θ 0 ] , g 2 = - ( γ 2 4 β + β 4 ) z + β c 3 2 δ g 1 + c 2 , g 3 = - [ γ / ( 2 β ) ] + c 3 f , θ 0 = { tan - 1 [ a 0 ( 4 β δ ) 1 / 2 ] } , c 2 = b 0 - a 0 c 3 2 ( β / δ ) , c 3 = ( c 0 + γ 2 β ) / f 0 ,
Γ s ( z , ρ ¯ c , ρ ¯ d ) = G s exp [ - a s ρ d 2 - b s ρ c 2 - i c s ρ ¯ c · ρ ¯ d ] ,
G s = ( G 0 b 0 ) / ( g 2 s ) , a s = g 1 s - [ ( g 3 s 2 ) / ( 4 g 2 s ) ] , b s = 1 / ( 4 g 2 s ) , c s = ( g 3 s ) / ( 2 g 2 s ) , g 1 s = δ z + a 0 , g 2 s = γ 2 δ z + γ 2 a 0 z 2 + c 0 γ z + b 0 , g 3 s = γ δ z 2 + 2 γ a 0 z + c 0 e ,
[ Γ s ( z , ρ c = W , ρ d = 0 ) ] / [ Γ s ( z , ρ c = ρ d = 0 ) ] = exp ( - 2 )
W c = ( 2 b s ) 1 / 2 = W 0 [ 1 + λ 2 z 2 π 2 W 0 4 ( 1 + 2 3 τ L W 0 2 l 2 ) ] 1 / 2 ,
2 3 τ L ( W 0 l ) 2 = 8 3 π 5 / 2 n 1 2 l - 1 z ( W 0 λ ) 2 1.
2 3 π 2 ( λ z l W 0 ) 2 τ L = 8 π 3 n 1 2 l - 1 z 3 W 0 2 1.
z max = ( 3 W 0 2 l 8 π n 1 2 ) 1 / 3 .
W f = ( λ z π W 0 ) [ 1 + 2 3 τ L ( W 0 l ) 2 ] 1 / 2 .
W f min = ( 2 3 τ L ) 1 / 2 ( λ z π l ) = ( 8 π 3 n 1 2 l - 1 z 3 ) 1 / 2 .
Γ = Γ 1 Γ 2 = Γ 1 exp [ - k d 2 8 A ( 0 ) z ] = Γ 1 exp [ - ½ ( ω d ω coh 2 ) 2 ] ,
ω coh 2 = v 0 ( π n 1 2 l z ) - 1 / 2 ,
η = ( ω d ω coh 1 ) 1 / 2 ;             ω coh 1 = 2 v 0 l π n 1 2 z 2 ,
π 4 n 1 2 ( z l ) 3 > 1.
π 3 / 2 n 1 2 ( z 2 l λ ) > 1.
P 0 ( ρ c , t - z v 0 ) = P i ( t ) G ( ρ c , t - z v 0 - t ) d t ,
G ( ρ c , t - z v 0 ) = 1 2 π ω d Γ ( z , ρ c , ρ d = 0 , ω d ) × exp [ - i ω d ( t - z / v 0 ) ] ,
Γ ( z , ρ c , ρ d = 0 , ω d ) = Γ 1 ( z , ρ c , ρ d = 0 , ω d ) Γ 2 = Γ 1 ( z , ρ c , ρ d = 0 , ω d ) exp [ - 1 2 ( ω d ω coh 2 ) 2 ] .
G ( ρ c , t - z v 0 ) = I 1 ( ρ c , t - z v 0 - t ) I 2 ( t ) d t ,
I 1 2 ( ρ c , t - z v 0 ) = 1 2 π Γ 1 2 ( z , ρ c , ρ d = 0 , ω d ) × exp [ - i ω d ( t - z / v 0 ) ] d ω d .
I 2 ( t - z v 0 ) = 1 ( 2 π ) 1 / 2 T 2 exp [ - ½ ( t - z / v 0 ) 2 T 2 2 ] ,
I 1 ( ρ c , t n ) = 1 2 π T 1 Γ 1 ( z , ρ c , ρ d = 0 , η ) exp ( - i t n η 2 ) d ( η 2 ) ,
P r ( t - z v 0 ) = P 0 ( ρ c , t - z v 0 ) A r ( ρ ¯ c ) d ρ ¯ c ,

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